Half-Plane of Absolute Convergence

221-

Extremal values of Dirichlet L-functions in the half-plane of absolute convergence

par JÖRN STEUDING

RÉSUMÉ. On démontre que, pour tout 03B8 réel, il existe une infinité de s = 03C3 + it avec 03C3 ~ 1 + et t ~ +~ tel que log log t log log log log t Re {exp i 03B8) log L (s, } ~ log

La démonstration est basée sur une version effective du théorème de Kronecker sur les approximations diophantiennes.

ABSTRACT. We prove that for any real 03B8 there are infinitely many values of s = 03C3 + it with 03C3 ~ 1 + and t ~ +~ such that Re { exp ( i 03B8) log L (s, } ~ log

The proof relies on an effective version of Kronecker’s approximation theorem.

1. Extremal values Extremal values of the Riemann zeta-function in the half-plane of absolute convergence were first studied by H. Bohr and Landau [1]. Their results rely essentially on the diophantine approximation theorems of Dirichlet and Kronecker. Whereas everything easily extends to Dirichlet series with real coefficients of one sign (see [7], §9.32) the question of general Dirichlet series is more delicate. In this paper we shall establish quantitative results for Dirichlet L-functions. Let q be a positive integer and let X be a Dirichlet character mod q. As usual, denote by s = Q + it with a, t E IEB, i2 = -1, a complex variable. Then the Dirichlet L-function associated to the character X is given by

- .

where the product is taken over all primes p; the Dirichlet series, and so the Euler product, converge absolutely in the half-plane 7 > 1. Denote by

Manuscrit regu le 9 juillet 2002. 222 xo the principal character mod q, i.e., Xo(n) = 1 for all n coprime with q. Then

Thus we may interpret the well-known Riemann zeta-function ((s) as the Dirichlet L-function to the principal character Xo mod 1. Furthermore, it follows that L(s, Xo) has a simple pole at s = 1 with residue 1. On the other side, any L(s, x) xo is regular at s = 1 with L(1, x) # 0 (by Dirichlet’s analytic class number-formula). Since L(s, X) is non-vanishing in Q > 1, we may define the logarithm (by choosing any one of the values of the logarithm). It is easily shown that for Q > 1

Obviously, IlogL(s,x)l::; L(a,xo) for a > l. However

Theorem 1.1. For any E > 0 and any real 0 there exists a sequence of s = a + it with a > 1 and t -> +oo such that

In particular,

In spite of the non-vanishing of L(s, X) the absolute value takes arbitrarily small values in the half-plane Q > 1! The proof follows the ideas of H. Bohr and Landau [1] (resp. [8], §8.6) with which they obtained similar results for the Riemann zeta-function (answering a question of Hilbert). However, they argued with Dirichlet’s homogeneous approximation theorem for growth estimates of 1((8)1] and with Kronecker’s inhomogeneous approximation theorem for its reciprocal. We will unify both approaches.

Proof. Using (2) we have for x > 2 223

Denote by cp(q) the number of prime residue classes mod q. Since the values x(p) are cp(q)-th roots of unity if p does not divide q, and equal to zero otherwise, there exist integers Ap (uniquely determined mod p(q)) with

Hence,

I I ’.A/ I In view of the unique prime factorization of the integers the logarithms of the prime numbers are linearly independent. Thus, Kronecker’s approximation theorem (see [8], §8.3, resp. Theorem 3.2 below) implies that for any given integer w and any x there exist a real number T > 0 and integers hp such that

Obviously, with w - oo we get infinitely many T with this property. It follows that

. , ,,¿- ,

provided that w > 4. Therefore, we deduce from (3)

/ I I I

resp.

in view of (2). Obviously, the appearing series converges. Thus, sending w and x to infinity gives the inequality of Theorem 1.1. By (1) we have

for a -t 1+. Therefore, with 0 = 0, resp. 0 = 7r, and a - 1+ the further assertions of the theorem follow. D

The same method applies to other Dirichlet series as well. For example, one can show that the Lerch zeta-function is unbounded in the half-plane 224 of absolute convergence:

if a > 0 is transcendental; note that in the case of transcendental a the Lerch zeta-function has zeros in Q > 1 (see [3] and [4]). In view of Theorem 1.1 we have to ask for quantitative estimates. Let 7r(x) count the prime numbers p x. By partial summation,

The prime number theorem implies for x > 2

By the second mean-value theorem,

for some ~ E (x, y) . Thus, substituting ~ by x and sending y - oo, we obtain the estimate

~ l-n

oo. This gives in (6)

Substituting (7) in formula (8) yields Re (exp(10) log L (a + iT, x)~

Let

then x tends to infinity as a - 1-~-. We obtain for x sufficiently large

The question is how the quantities w, x and T depend on each other. 225

2. Effective approximation ° H. Bohr and Landau [2] (resp. [8], §8.8) proved the existence of a T with 0 T exp(N6) such that

where p, denotes the v-th prime number. This can be seen as a first effective version of Kronecker’s approximation theorem, with a bound for T (similar to the one in Dirichlet’s approximation theorem). In view of (5) this yields, in addition with the easier case of bounding ,(( s) from below, the existence of infinite sequences S:i: = a± + it± with cr~ 2013~1+ +oo for which

where A > 0 is an absolute constant. However, for Dirichlet L-functions we need a more general effective version of Kronecker’s approximation theorem. Using the idea of Bohr and Landau in addition with Baker’s estimate for linear forms, Rieger [6] proved the remarkable

Theorem 2.1. Let v, N E N, b E ~,1 w, U E R. Let pi ... PN be prirrae numbers (not necessarily consecutive) and

Then there exist hv E Z, 0 v N, and an effectively computable number C = C (N, PN) > 0, depending on N and pN only, with

We need C explicitly. Therefore we shall give a sketch of Rieger’s proof and add in the crucial step a result on an explicit lower bound for linear forms in logarithms due to Waldschmidt ~9~. Let K be a number field of degree D over Q and denote by the set of logarithms of the elements of {0}, i.e.,

If a is an algebraic number with minimal polynomial P(X) over Z, then define the absolute logarithmic height of a by

I -l1

note that h(a) = loga for integers a > 2. Waldschmidt proved 226

Theorem 2.2. E LK and /3v for v = 1, ... , N, not all equal zero. Define av = for v = 1, ... , N and

Let E, W and v N, be positive real numbers, satisfying

and

Finally,, define Vv = 1} for v = N and v = N - 1, with Y+ = 1 in the case N = l. If A ~ 0, then

with

This leads to

Theorem 2.3. With the notation of Theorem 2.1 and under its assump- tions there exists an integer ho such that (11) holds and

if PN is the N-th prime number, then, for any E > 0 and N sufficiently large,

Proof. For t define 227

With > we have

By the multinomial theorem,

Hence, for 0 B 6 R and kEN

By the theorem of Lindemann

vanishes if and only if jv = jv for v = -1, 0, ... , N. Thus, integration gives

, and

if jv for some v E {20131,0,..., N~ . In the latter case there exists by Baker’s estimate for linear forms an effectively computable constant A such that _ 228

Setting have, with the notation of Theorem 2.2,

= = = = for v N. We may take E = 1, W logpjv, Vi 1 and Vv log pv 2, ... , If N > 2, Theorem 2.2 gives ...T ....

I Thus we may take

Hence, we obtain

Since

application of the Cauchy Schwarz-inequality to the first multiple sum and of the multinomial theorem to the second multiple sum on the right hand side of (16) yields I B 2

Setting A defined by

we obtain 229

This gives

note that By definition 7BJ

Therefore, using the triangle inequality, and arbitrary t E 118. Thus, in view of (17)

If hv denotes the nearest integer to then

For v = 0 this implies 17 - ho~ G ~. Replacing T by ho yields

Putting we get

Substituting (15) and replacing b-1 by b, the assertion of Theorem 2.1 follows with the estimate (12) of Theorem 2.3; (13) can be proved by standard estimates. D

3. Quantitative results We continue with inequality (9). Let pnr be the N-th prime. Then, using Theorem 2.3 with N = 7r(x), v = u" = 1, and

yields the existence of T = 27rho with

such that (4) holds, as N and x tend to infinity. We choose cv = log log ~, then the prime number theorem and (18) imply log x = log N + 0 (log log N), log N > log log log T + 0 (log log log log T) . Substituting this in (9) we obtain 230

Theorem 3.1. For any real 0 there are infinitely many values of s = a+ it with a - 1-~ and t -> --~oo such that

Using the Phragm6n-Lindel6f principle, it is even possible to get quantitative estimates on the abscissa of absolute convergence. We write f (x) _ with a positive function g(x) if

hence, = is the negation of f (x) = o(g(x)). Then, by the same reasoning as in [8], §8.4, we deduce

and

However, the method of Ramachandra [5] yields better results. As for the Riemann zeta-function (10) it can be shown that and further that, assuming Riemann’s hypothesis, this is the right order (similar to (8~, §14.8). Hence, it is natural to expect that also in the half- plane of absolute convergence for Dirichlet L-functions similar growth estimates as for the Riemann zeta-function (10) should hold. We give a heuristical argument. Weyl improved Kronecker’s approximation theorem by

Theorem 3.2. Let ai , ... , aN E II8 be linearly independent over the field of rational numbers, and let 7 be a subregion of the N-dimensional unit cube with Jordan volume r. Then 1

Since the limit does not depend on translations of the set y, we do not expect any deep influence of the inhomogeneous part to our approximation problem (4) (though it is a question of the speed of convergence). Thus, we may conjecture that we can find a suitable T exp(N’) with some positive constant c instead of (13), as in Dirichlet’s horraogeneous approximation theorem. This would lead to estimates similar to (10). 231

We conclude with some observations on the density of extremal values of log L(s, X). First of all note that if holds for a subset of values T E [T, 2T] of measure pT, where f (T) is any function which tends with T to infinity, then

In view of well-known mean-value formulae we have fl = 0, which implies

This shows that the set on which extremal values are taken is rather thin. The situation is different for fixed Q > 1. Let Q be the smallest prime p for which 0. Then

note that the right hand side tends to 0+ as o- --~ +oo, and that Q q +1.

Theorem 3.3. Let 0 6 1. Then, for arbitrary 0 and fixed a > 1,

Proof. We omit the details. First, we may replace (2) by

This gives with regard to (8)

for some integer ho = m, satisfying (12), where N = 7r (x) and cos 2w = 1 - J - Putting x = Q2, proves (after some simple computation) the theorem. 0 For example, if X is a character with odd modulus q, then the quantity of Theorem 3.3 is bounded below by 232

Acknoycrledgement. The author would like to thank Prof. A. Laurincikas, Prof. G.J. Rieger and Prof. W. Schwarz for their constant interest and encouragement. References

[1] H. BOHR, E. LANDAU, Über das Verhalten von 03B6(s) und 03B6(k)(s) in der Nähe der Geraden 03C3 = 1. Nachr. Ges. Wiss. Göttingen Math. Phys. Kl. (1910), 303-330. [2] H. BOHR, E. LANDAU, Nachtrag zu unseren Abhandlungen aus den Jahren 1910 und 1923. Nachr. Ges. Wiss. Göttingen Math. Phys. Kl. (1924), 168-172. [3] H. DAVENPORT, H. HEILBRONN, On the zeros of certain Dirichlet series I, II. J. London Math. Soc. 11 (1936), 181-185, 307-312. [4] R. GARUNK0160TIS, On zeros of the Lerch zeta-function II. Probability Theory and Mathemat- ical Statistics: Proceedings of the Seventh Vilnius Conf. (1998), B.Grigelionis et al. (Eds.), TEV/Vilnius, VSP/Utrecht, 1999, 267-276. [5] K. RAMACHANDRA, On the frequency of Titchmarsh’s phenomenon for 03B6(s) - VII. Ann. Acad. Sci. Fennicae 14 (1989), 27-40. [6] G.J. RIEGER, Effective simultaneous approximation of complex numbers by conjugate algebraic integers. Acta Arith. 63 (1993), 325-334. [7] E.C. TITCHMARSH, The theory of functions. Oxford University Press, 1939 2nd ed. [8] E.C. TITCHMARSH, The theory of the Riemann zeta-function. Oxford University Press, 1986 2nd ed. [9] M. WALDSCHMIDT, A lower bound for linear forms in logarithms. Acta Arith. 37 (1980), 257-283. [10] H. WEYL, Über ein Problem aus dem Gebiete der diophantischen Approximation. Göttinger Nachrichten (1914), 234-244.

Jorn STEUDING Institut fur Algebra und Geometrie Fachbereich Mathematik Johann Wolfgang Goethe-Universitat Frankfurt Robert-Mayer-Str. 10 60054 Frankfurt, Germany E-maiL : steuding0math.uni-irankfurt.de